The contest was decided from the spot after Musa Bilankulu cancelled out new Sundowns signing Fiston Abdul Razak's 105th minute opener at Lucas Moripe Stadium.
Celtic - who joined holders Kaizer Chiefs, Bidvest Wits and Ajax Cape Town in the next round - slotted all of their penalties, while Asavela Mbekile and Kennedy Mweene failed to do so for the Brazilians.
The opening half was a cagey affair, although Downs recruit Keagan Dolly tested Patrick Tignyemb with a useful effort just past the half-hour mark.
Dolly then teed up Razak before the break but the Burundi international ballooned his attempt.
While Phunya Sele Sele threatened after the interval through efforts from Musa Nyatama and Keagan Buchanan, Dolly came closest to breaking the deadlock in regulation time when his curling free-kick struck the upright in the 73rd minute.
Razak eventually drew first blood when he volleyed home from close range following a pass from Zimbabwean strike partner Cuthbert Malajila.
However, Bilankulu equalised in the 118th minute when he nodded in a cross from Lyle Lakay to force the game to penalties.
The visitors ultimately held their nerve, with Bevan Fransman slotting the winning kick for Clinton Larsen's outfit.